Paper received: 10.04.2009 Paper accepted: 07.08.2009 Uniform Power Transmission Gears Gorazd Hlebanja* - Jože Hlebanja University of Ljubljana, Faculty of Mechanical Engineering A new type of lubricated gears is presented in this paper, referred to as Uniform Power Transmission Gears (UPTG), which have conformal shaped teeth flanks that enable a more uniform transmission of power and motion. The proposed teeth flanks comprise three arcs: an addendum arc, a dedendum arc, with both of these joined by a connecting arc. These UPT gears have been derived from Hawkins' patented Zero Sliding Gears (ZSG) [4]. The name Uniform Power Transmission Gears derives from the fact that these gears operate during power transmission with an almost constant value of friction and a nearly constant value of power transmission. Thus, the power transmission of the UPT gears occurs simultaneously with the double-contact teeth flanks of the driving and driven gears, which means the transmitted power is divided into two parts: the contact pressure in the contacts is lower, and the gear pair conjugates as helical gears. UPT gears can be produced on common gear cutting machines based on a rolling process where teeth shapes are formed by successive enveloping cuts of a cutter. However, the cutter teeth should be manufactured by the UPT rack profile. A comparison of the UPTGs with involute-type gears is presented. The radii of curvature are investigated in detail and the velocities in the teeth flank contacts were analyzed. Additionally, an investigation of oil-film thickness using the theory of Hamrock and Dowson is discussed. Owing to the decisive influence of heat caused by friction on gears' scuffing resistance, and the heating of the teeth flanks based on Blok's flash-temperature criterion, we have made a careful study and the results are presented. Because the power transmission with UPTGs occurs mostly by rolling, with only a minor amount by sliding, such gears are very suitable for heavy-duty power transmission at relatively slow speed working conditions. © 2009 Journal of Mechanical Engineering. All rights reserved. Keywords: zero sliding gears; conformal helical gears; oil-film thickness; flash temperature; uniform power transmission gears 0 INTRODUCTION In terms of efficiency, for high-speed applications such as the gear drives of gas turbines, which have high contact velocities and thick oil films based on low viscosity oils, there is little likelihood of making significant improvements to the existing involute technology. However, in low-speed industrial applications it is still possible to improve the efficiency and reduce the manufacturing costs. By making the gears conformal we make it possible to have a many-fold increase in the thickness of the oil film [1]. This means that applications such as windturbine gearboxes and winch-gear reducers, which rely on heavy oils to provide an adequate oil-film thickness, could use lighter oils to reduce friction while maintaining the same oil-film thickness. The combination of using lighter oils and a tooth form that has a reduced specific sliding would improve the overall gear-mesh efficiency in these applications. Furthermore, the energy savings can be shown to be substantial over the lifetime of the gear drive. Smaller contact pressures and reduced mesh losses also allow for a reduction in the size of gears and the cost of the related oil-cooling system. Manufacturing costs are also an important consideration. Many applications are using evermore-expensive gearings. For example, carburized and ground gears, which although they provide a high level of performance are also very costly. The use of conformal gearings, providing the gears are made to be sufficiently conformal, allows hardened gears to be substituted with less-costly materials while staying within the same size envelope. In 2007, Hohn, et al. [2] reported on redesigned involute gearing and gear-mesh losses reduction by as much as 68%. This provides *Corr. Author's Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, gorazd.hlebanja@fs.uni-lj.si conclusive proof that it is possible to design involute gearings with improved efficiency. The basic technique that they employed was to go to a higher pressure angle, use a shorter tooth height and a lower profile contact ratio, and make wider gears with a helical contact that is sufficient to overcome the reduction in the profile contact ratio. Other significant modifications are given in more detail in [2]. Another, alternative approach that can improve efficiency without the need to resort to wider gears is to consider specific types of non-involute gears [3]. withdrawn areas of tooth flank working areas of tooth flank Fig. 1. Hawkins' zero-sliding gears [9] 1 HAWKINS' ZERO SLIDING GEARS Zero sliding gears were granted a US patent in January 2005 [4], and their principle of operation is shown in the photograph in Fig. 1. Hawkins proposed to withdraw a part of the tooth flank relieving central portion of the S gear flanks, thereby eliminating the contacts between teeth flanks in transverse plane of gears. Such gears should be of the helical type so as to enable the transmission of power and the uniform rotation of the driving and the driven gears. During power transmission with these gears, the contact areas of the teeth flanks roll in the axial direction and slide around the pitch point. The addendum and the dedendum of these teeth flanks are shaped by two cycloidal arcs and are separated by the withdrawn parts of the flank. The power transmission from the driving gear to the driven gear occurs over the contact points of the two pairs of teeth, but the first contact is always from the dedendum tooth flank of the driving gear to the addendum tooth flank of the driven gear, with the sliding in the transverse plane and the rolling between the contacting surfaces of the helical lines being in the axial direction. In the second contact the power transmission occurs in the same way, but with reversed relative positions of the teeth flanks. All of the contacts of the gear flanks roll from the front side of the gears to the back side, or vice versa, depending on the direction of the helix. New partners come to contact after every pitch length. Hawkins describes how these ZSGs, hardened and induction-case-hardened gears, were rig tested without any problems being observed. There was no excessive noise or UPT profile (right) Detail A UPT straight line DF' through the pitch point C (involute shaping line) arc 12, the shapable part of a tooth flank declination X_ of a tooth flank Fig. 3. Theoretical shape of the rack for the basic UPT teeth profile vibration, and no evidence of distress. However, titanium versions of the gears only worked for as long as the coating remained intact; once the coating wore off the gears tended to fail. 2 THE DESIGN OF A NEW RACK PROFILE The photograph in Fig. 1 shows that the profiles of the ZSG teeth are relatively slim and sharpen to a point. This means that these ZSGs are, more or less, only of theoretical interest and are certainly not suitable for heavy-duty gear applications. It is this observation that led to the idea of developing a new gear type, with the aim to optimize the contact area of the conformal tooth flanks. This new type of gears will be presented and discussed in the paper. We will start with the tooth flank profile for a gear with z = to, the rack profile, illustrated in Fig. 2. The point S is the centre of the addendum and dedendum circles of the tooth flanks. The line DF with the slope angle a = 15o (a could also vary in either direction) is chosen to estimate the radii of these circles. In order to reduce the sensitivity of the gears to deviations in the gears' centre-to-centre distance (from S to S'), radii of the tooth flank profiles for the radius of the addendum ra and for the radius of the dedendum rd have been made different. However, the sum of both radii should be kept constant: ra + rd = mncosa = p . (1) According to this equation, the addendum radius is ra = mnkcosa , and the dedendum radius rd = mn(1 -k)cosa , where the factor k < 0.5, and with which the difference between both radii, rd - ra, is estimated. The product mn is the circular pitch p of the gears, and the product mucosa is the distance between two neighbouring pairs of teeth contacts. The arc DE from circle ra, and the arc FG from the circle rd define the shape of the tooth addendum and the tooth dedendum parts of the teeth flanks of the gears with z = to. Since the rack cutter is also a gear with the pitch radius R = to, the teeth flanks are equal, only the tooth spacing and the tooth thickness are changed. Then we shift the arc FG to the left by the length of the circular pitch p = mn to the position F'G' and connect both arcs with a new arc 12, which is part of the circle s with the radius R. In this way the whole tooth profile of the gear with z = to or of the rack cutter is defined. If a smooth profile for the tooth flank is required, then point 1 should be placed on the connecting line OS and point 2, on the connecting line OS'. The UPT gears' tooth flank profile of the rack cutter is, therefore, established by the curve E-D-1-2-G'. With the profile presented in Fig. 3, a new type of rack cutter is used for manufacturing the UPT gears. This is composed of a left- and right-handed tooth profile, connected by fillet arcs with the fillet radius p. With such tools the gears can be manufactured with a rolling process, where the teeth formed on the work-piece are shaped by successive enveloping cuts, and the cutting profiles of the rack correspond to this theoretically shaped, basic rack profile. With regard to an arbitrary gear based power transmission, UPT gears experience backlash, and, therefore, the tooth thickness, s, should be smaller than its space width, e. In our dedendum circle adddendum circle Fig. 4. UPT rack profile case the backlash, j, equates to double the difference of the radii rd— ra, i.e., j = 2(rd — ra). In the transverse plane, the UPT gears are only able to operate with the cylindrical parts of the teeth flanks. This is illustrated in detail A in Fig. 3. The arc 12 is inserted between the arcs E1 and 2G (see Fig. 2) of the UPT flank profile, and, therefore, in the region of the rack's datum line the rack-tooth thickness increases by an amount X on each side. For this reason each successive cut by the enveloping gear tooth becomes increasingly smaller at this point, and, therefore, such gears cannot operate (as observed in the transverse plane) in that part of the tooth flank, but only in dedendum and addendum parts of the gear teeth flanks as helical gears. The power transmission with UPT gears is shown in Fig.4, i.e., always through the diametrically opposite contacts Pa, for the addendum circle, and Pd, for the dedendum circle, on the tooth flanks. These contact points are situated approximately in the middle of the arcs ED1 and F'G' (see Fig. 2), and are located on the connected line PaSPd . A circle, named the sliding circle, passes through both contact points, with its centre, C, being equidistant between Pa and Pd. The centre, C, the pitch point, also lies on the datum line (or the pitch line). The direction Pa CPd is also the direction of power action. The rack cutter could be designed for the transverse plane, as illustrated in Fig. 4, or for the normal section. In order to apply the same cutting tool a chosen arbitrary module and another helix angle, it would be reasonable to design a cutting tool for the normal section. 3 GENERATION OF UPT GEARS WITH A UPT RACK PROFILE UPT gears are manufactured by a generating process where the rack-shaped tool (the rack-type cutter, the hob, etc.) with its datum line rolls in the mesh with the gears' pitch circle without sliding, and the rack cutter's teeth flanks form the gear teeth by successive enveloping cuts, as illustrated in Fig. 5. The shape of the cutter's (the hob, the rack cutter, the pinion cutter, the milling head, the grinding tool, etc.) tooth profile should conform to the one defined in Figs. 3 and 4, and to the shape described in Section 3. The main features are the addendum and dedendum circles, which are bound to the datum line of the rack cutter, which means the addendum of the Fig. 5. Generating UPT gears for any appliance with a UPT profiled cutter ¿datum line driven gear rear side I Fig. 6. Gear teeth generated by the rolling of the rack profile in the mesh over the pitch circle Fig. 7. Pair of UPT gears with a double concave-convex contact of teeth gear tooth becomes the dedendum circle of the rack profile and the dedendum of the gear tooth becomes the addendum circle of the rack profile. All the gears that are produced with the same shape of rack get teeth with the same radius of addendum ra, the same radius of dedendum rd, and the same helix angle p, which implies that all the gears with the same module and the same helix angle p can be manufactured with the single tool. The two gears represented in Fig. 6, i.e., z1 = 20 and z2 = 30, are designed with the same rack profile and can work together, as shown in Fig. 7. front side Fig. 8. Driven and driving UPT gears in axial view J contact area on the driving gear ^ contact area on the driven gear Fig. 9. UPTgear pair in working position There are two concave-convex contacts: the first is at point Pd, where the power transmission between the dedendum of the driving gears and the addendum of the driven gears occurs, and the second, at point Pa, where the transmission from the addendum of the driving gear to the dedendum of the driven gear occurs. It is important to point out that the gear's teeth are bent in the axial direction, as illustrated in Fig. 8, where the driving and driven gears are presented separately. In the working position both gears should be positioned so that the pitch points C coincide, as schematically shown in Fig. 9. The mutually interacting surfaces are marked and they come into contact at Pa and Pd in the transverse plane, positioned at the pitch point, as exposed in Fig. 9, where the gears are shown in the working position and in the axial view. 4 COMPARISON OF THE INVOLUTE WITH THE UPT TOOTH PROFILE UPT gears are functional only as helical gears; therefore, they are comparable with helical involute gears. Such a comparison of the tooth shape of involute gears (E-gears) with the tooth shape of UPT gears, in the normal section, is illustrated in Fig. 10 for gears of the same module, the same number of teeth, the same top circle, the same root diameter and the same tooth thickness, 5, on the pitch circle. For the tooth profile of the involute gear from Fig. 10a the usable involute tooth profile ends on the base circle. The gears' bending stress depends on the tooth thickness of the dedendum Sf and the height hf of the load-crossing line with the mid-line of the tooth profile. The corresponding UPT tooth profile is shown in Fig. 10b. The UPT tooth profile is involved in power transmission, both at the addendum and at the dedendum part. It is also evident that the altitude of the load-crossing height is smaller and the tooth thickness of the dedendum Sf is greater to some extent. Fig. 10c illustrates a superposition of both profiles. In comparison to the UPT teeth, the involute teeth are thicker in the pitch circle area and weakened at the bottom area. In terms of the bending strength, the UPT gears are stronger of the two. 5 THE TRANSMISSION OF POWER AND MOTION WITH UPT GEARS The power transmission between a UPT driving gear and a UPT driven gear is illustrated in Figs. 9, 11 and 12. An important feature of the UPT gears, which can only operate as helical gears, is that they do not possess the normal path of contact in the transverse plane, as depicted in Fig. 11. This figure shows that the teeth flanks move from the working position at A, pass the pitch point C, to the working position B, without any contact between the flanks, and consequently, also without any friction. From Fig. 12 it can be concluded that the power can only be transmitted over the contacts of the simultaneously loaded surfaces at Pa and at Pd. Both contact spots converge to points on two pairs of helices, one on the pinion and the other on the gear in each pair. The contacts Pa and Pd also represent tangential points of the pinion and the gear helices. From the transverse view in Fig. 12 the rotation and the power transmission from the pinion to the gear with the sliding of the pinion flank surfaces along the gear flanks can be recognised. At the contact point Pd the pinion's dedendum flank slides along the gear's addendum, and at point Pa the pinion's addendum flank slides in the opposite direction across the gear's dedendum flank with velocity vg. The pinion's helices through points Pa and Pd rotate with the pinion and the gear's helices in the same contact points rotate with the gear. Therefore, also the contact points, that is Pa and Pd are moving in the axial direction by rolling the pinions' helices over the gears' helices in the direction from the pinion and gear front side to their back side. 5.1 The Velocities of the Teeth Flank Contacts As mentioned above, the same teeth flanks that were in contact at point Pd, came by rotation of the pinion around its axis O1 and the rotation of the gear around the axis O2 to the contact at point Pa, which means that the rotation of the pinion and the gear pitch circles for an arc size of one pitch p, and at the same time the contact points Pd and Pa shift from the front side to the back side of the pinion (or gear) for the length of the face -—K PJ&circie Fig. 10. (a) The involute tooth profile; (b) the UPT tooth profile, both in normal section; (c) comparison of the involute and UPT shapes driven left tooth flank Fig. 11. Progress of the teeth flanks without contact in the transverse plane from the contact point A to the contact point B during power transmission contact start-P; gear and pinion front side Fig. 12. Velocities at the teeth contact for the UPT gears width b with the rolling velocity vr. If the pinion rotates with a constant angular speed co1, then the gear rotates with a constant angular speed co2, and then the pitch line velocity vt for both the pinion with the pinion pitch radius r01 and for the gears with the pitch radius r02, equals: vt=ai roi=a2 ro2. (2) If the angular velocities ra1 and ra2 remain constant over time, then the pitch line velocity vt will also remain constant for the same interval. At the same time, as the pitch circle passes the length t - torsional radii of helices Fig. 13. Load transmission in the UPT gears of the transverse pitch p, the contact points Pa and Pd should be shifted by the face width b, where b = p/tan p, (P is the helix angle). From this relation derives the axial velocity of the contact points Pa and Pd in the axial direction from the front side to the back side, vr: /tanß. (3) The relative angular velocity rar of the pinion and gear around the pitch point is defined as ar= a>1+ a>2. (4) The sliding velocity, with rs = 0.5 (m n cos a), between the teeth flanks vg is Vg=rs(a>1+ ®2) = rsar (5) Thus, the angular velocity ar and the sliding velocity vg are proportional. It is important to point out that in case of constant angular velocity ar also the sliding velocity vg remains constant. 5.2 Power Transmission Between the Teeth Flanks The power transmission for all the gears starts with the driving shaft and the input torque, T1 in, which transmits the rotation over the gear teeth with the tangential force Ft to the driven, output shaft with torque T2out. In this way the power is also transmitted with UPT gears, for which a tangential force Ft, a radial force Fr and an axial force Fa with its resultant Fn as a normal v r addendum tooth flank (gear) dedendum tooth flank (pinion) gear's heb_ ^contact'area. Fig. 14. Shape of the contact area Pd force are presented in Fig. 13. From the known input torque T1 it is possible to derive the tangential force (providing the losses are neglected): Ft = Ti/rOT. (6) The normal force Fn acts perpendicularly to the teeth flanks and creates an axial component Fa and a radial component Fr. Therefore, the axial component Fa operates along the gear axis, Fa = Ft.tan/, while the radial component Fr operates perpendicularly to the axis Fr = Ft.tana. In this way the normal force Fn, the force with which the teeth flanks are pressed against each other, is obtained by taking the square root of the sum of the square of all its components: Fn = (Ft + F2 + F ) = 0.5 Fn (1 + tan ß2 + tana2)05. (7) Fig. 13 shows that the normal force Fn operates through both contact points Pa and Pd as it does through the pitch point C. Therefore, the contact points Pa and Pd under working conditions in the axial direction fall behind each other for a certain distance: 5 = mn cosa sin/ (8) The input torque T1 usually remains constant for some time, which is consequently also true for the normal force Fn. This then implies that the power transmission with UPT gears also stays constant during this time period. It is for this reason that these gears have been named Uniform Power Transmission gears, or UPT gears, in short. 5.3 The Shape of the Contacts of the Teeth Flanks The contacts over which the power transmission occurs are composed of the driving part and the driven part where the driving part presses the driven part over the contact area, resulting in a sliding or rolling motion. The contact areas of the UPT gears are ellipsoidal in shape; this is illustrated by the contact point Pd shown in Fig. 14. The driving pinion with its dedendum part of the tooth, drives the addendum part of the gear tooth, the contact area then rolls along the gear's helix with the velocity vk, but simultaneously the dedendum flank slides over the addendum flank with the velocity vg. The axis of the elliptically shaped contact can be defined by the radii of curvature of the helices zdl and zdz, as well as by the radii of the tooth flanks ra and rd, as shown in Fig. 4 for the contact point Pd. The tooth flank addendum radius ra and the dedendum radius rd are equal for both contact points Pa and Pd. However, the torsional radii of the helices rd1 and rd1 differ in these contact points. The helices are located on the cylinder surfaces with the radius rP1 for the pinion and rP2 for the gear at the contact point Pd. The torsional radii of helices can be calculated with the formula [8], [10] z=(ar2+br2)/b, (9) where ar equals the radius of the cylinder surface rP and br = rP tan/?. Taking into account Eq. (9), the torsional radii of the helices with the course through the contact point Pd are: rdi - rpi + (rPitan ß)2 , and Td 2 - rP1 tan ß rp2 + (rp2 tan ß)2 (10) rP2tan P The reduced radius of curvature for the rolling contact point Pd is defined by: (11) 1 _ _ _1_++ Pred,k ra rd Td1 Td 2 whereas the value for the reduced radius of curvature for the sliding contact is: 1 1 1 Pred ,s (12) Based on the above explanation, a concave-convex type of contact should exist between the teeth flanks and a convex-convex r r d type of contact between the helices. In order to illustrate the contact circumstances, some data for gears in Figs. 6 and 7, with m = 25 mm, are presented, such as radii ra = 37.5 mm, rd = 41.23 mm, Tfl = 472 mm, and rd2 = 783 mm. The reduced radius for the contact point Pd is thus pred,k = 168 mm. A similar result for the reduced radius of curvature can also be expected for the contact point Pa, where the power transmission occurs from the addendum flank of the pinion to the dedendum flank of the gear. To compare the reduced radii of curvature of the UPT gears to the involute gears, a reduced radius for the involute gear pair of the same size (z1 = 20, z2 = 30, m = 25 mm) was calculated, and pred,inv = 50.8 mm has been obtained as the maximum value. 5.4 Hertzian Contact Stress The most commonly used criterion for the pitting durability on gear-tooth flanks is the Hertzian contact pressure. For involute gears a rectangular contact area, A = 2b l, has to be taken into account, where b stands for half the contact width, l is face width, and the maximum pressure is determined with the following formula [6]: Fn E r, w Er, lim ' , ..........(13) 1 2nl Pred \ 2n Pred For a given tangential load Ft for involute gears, the contact load is Fn = Ft /cosa, where a = 20o and Fn = 1.064 Ft. For good operating conditions the Hertzian pressure is limited by the allowable pressure aiim, as well as the reduced radius of curvature pred and the specific load w = Fn/l, which also have a strong influence on the pressure. According to Fig. 14, an elliptical contact area exists for the UPT gears where the contact pressure can be expressed by the formula [7]: Fn *H = ^ ' (14) where the expressions for a and b are: 1/3 a = kl 3Fn Pred I , and E red b = k 3Fn pred E, red 1/3 (15) from which the following formula is obtained: Fig. 15. Sliding and rolling directions of the Pd contact area 3 F 2 E , M n red